11 THE SPECTRAL THEORY OF HUBERT 405 (f) Let (An) be the sequence of eigenvalues of a nuclear operator Ty each counted according to its algebraic multiplicity (11.4.1). Show that]T \Xn\ rg \\T\\i, (For each n integer^, consider the sum V of the p spaces N(JU*; T) (I <J k <*p), where filt..., fj.p are the first p distinct eigenvalues in the sequence (An). Take a Hilbert basis of V with respect to which the matrix of T\ V is triangular, and use (c).) Deduce that if T is a Hilbert-Schmidt operator and if (An) is the sequence of its eigenvalues, each counted with its algebraic multiplicity, then]T |An|2 <£ \\T\\Z- n (g) If a continuous operator Te J*?(E) is such that, for each pair of Hilbert bases (an\ (bn) of E, the series £ (T * an \ bn) is convergent, then T is nuclear. (Write n T = LU* (Problem 6(a)), and by choosing (an) and (bn) suitably show that Llf2 is a Hilbert-Schmidt operator.) (h) For a continuous operator Te &(E) to be nuclear, it is necessary and sufficient that, for at least one Hilbert basis (<?) of E, the seriesX \\T- en\\ should converge. (Write T= UR (Problem 6(a)) and note that (R-en\ en) <; || T - en\\. This proves that the condition is sufficient. Conversely, take for (en) a basis consisting of eigenvectors of R.) (i) In the space E = /£, let (en) be the canonical Hilbert basis, and let a 1L OA*Xi- If F is the subspace C a of dimension 1, show that the projector PF n is nuclear, but that the series]T \\PF - en\\ does not converge. n 8. Show that, for every normal operator W on a separable Hilbert space E, there exists a normal operator N' such that N'2 = TV. Give examples where there exist infinitely many such operators, 9. Let T be a continuous operator on a separable Hilbert space E. (a) For T to be a topological left zero-divisor (Section 15.2, Problem 3) in the algebra ^f(E), it is necessary and sufficient that there should exist a sequence (xn) of vectors in E such that \\xn\\ = 1 for all n and such that (T xn) tends to 0. The complex numbers £ such that T £ 1 is a topological left zero-divisor in ^(E) form what is called the approximative point-spectrum Spa(T). Thus £ $ Spa(3n) means that T £ 1 is injective and a horneomorphism of E onto a closed subspace of E. Show that Spa(T) is closed in C, and contains the frontier of Sp(!T). If P is any polynomial, show that Spa(P(D)-P(Spa(D). (b) LetTe JS?(E). For each A e C, let m(T, A) denote the dimension of Ker(r*- A 1) (so that m(T, A) is either a nonnegative integer or + oo), equal also to the codimension of (T A 1)(E), Let r0 e £?(E) and let A0 be a complex number not belonging to Spa(To). Show that there exists a number e>0 such that m(T, A) = m(TQ, A0) whenever \\T- TQ\\ <; e and |A - A0| g e. (c) Deduce from (b) that if K is a compact subset of C which does not intersect Spa(J'o), then there exists e>0 such that w(rğA) m(r0,A) for all AeK and all Te &(E) such that \\T-TQ\\<: e. (d) Let T e &(E) and let K be a compact subset of Sp(7) - Spa(T) with the follow- ing properties: (1) 0 $K; (2) the inverse image K/ of K under the mapping £iğ£2 is convex; (3) w(r,A)= 1 for all AeK. Then there exists no operator T'e &(E) such that T/2T, (Supposing the contrary, let L = K/nSp(r/); then we have L <= Sp(r) - Spa(TO, and L u (-L) = K'. Show that A e L implies that ~A f Low that the function